Photometry

Measuring Energy

• Photometry measures the energy from a source using a narrow range of wavelengths.

– Visual wavelengths from 400-700 nm

– Narrower slice of wavelengths

• Photometry uses filters to select wavelengths.

• Spectroscopy measures energy over a wide range of wavelengths.

– Visual spectrum

– UV, IR spectra

– Full EM spectra

• Spectroscopy requires instruments to get at each wavelength separately.

– Interferometer

Luminosity of Stars

• Luminosity measures how much energy is produced.

– Absolute brightness L

• Relative luminosity is usually based on the Sun.

• Astronomers measure luminosity relative to the Sun.

– LSun = 1 L

– LSirius = 23 L

• Stars range from 0.0001 L to 1,000,000 L .

Magnitude

• The observed brightness is related to the energy received.

• The magnitude scale was originally 6 classes.– Effectively logarithmic

• The magnitude (m) was made formal in 1856.– Lower numbers brighter– 6m at the limit of human

vision

)/log(5.2 mn EEnm

512.210 5.21 m

n

E

E

For 1 unit of magnitude:

Brightness Magnified

• Images from a telescope must fit within the pupil.

– Brightness proportional to the aperture squared

– Ratio of observed to natural

• No increase for extended objects from magnification.

– Eg. M31(> moon)

– Light on more rods

– Exclusion of other light

Df

fP

o

ee

o

f

fM

1

2

2

P

MD

L

LR

eye

telescope

Point Source Magnified

• Point sources are smaller than one pixel (or rod).– No increase in image size

from magnification

• The ratio of brightness increase is the light grasp G.– Pupil size 7 mm

• The limiting magnitude comes from the aperture.– CCD 5 to 10 magnitudes

better

Dm 10min log58.16

224 )(m102 DG

2

2

P

DG

in meters

8” aperture is 13.3m

Apparent Magnitude

• The observed magnitude depends on the distance to the source.

– Measured as apparent magnitude.

• The scale is calibrated by stars within 2° of the north celestial pole.

• Some bright stars (app. mag.):– Sun -26.7– Sirius -1.4– Alpha Centauri -0.3– Capella 0.1– Rigel 0.1– Betelgeuse 0.5– Aldebaran

0.9

• These are all brighter than first magnitude (m = 1.0)

Distance Correction

• Brightness falls off as the square of the distance d.

• Absolute magnitude M recalculates the brightness as if the object was 10 pc away.

– 1 pc = 3 x 1016 m = 3.26 ly

• The absolute magnitude can be corrected for interstellar absorption AD.

100log5.2

2dmM

dmM log55

ADdmM log55

AD = 0.002 m/pc in galactic plane

Absolute Magnitude

• Distance is important to determine actual brightness.

• Example: 2 identical stars

A is 7 pc, B is 70 pc from Earth

The apparent brightness of B is 1/100 that of A

The magnitude of B is 5 larger.

• Some bright stars (abs. mag.):– Sun 4.8– Sirius 1.4– Alpha Centauri 4.1– Capella 0.4

– Rigel -7.1– Betelgeuse -5.6– Aldebaran -

0.3

• These are quite different than their apparent magnitudes.

Imaging

• Photographic images used the width of an image to determine intensity.

– Calibrate with known stars

– Fit to curve

• CCDs can directly integrate the photoelectrons to get the intensity.

– Sum pixels covered by image

– Subtract intensity of nearby dark sky

• Data is corrected for reddening due to magnitude and zenith angle.

IBAD 10log

Solar Facts

• Radius:

– R = 7 105 km = 109 RE

• Mass :

– M = 2 1030 kg

– M = 333,000 ME

• Density:

– = 1.4 g/cm3

– (water is 1.0 g/cm3, Earth is 5.6 g/cm3)

• Composition:

– Mostly H and He

• Temperature:

– Surface is 5,770 K

– Core is 15,600,000 K

• Power:

– 4 1026 W

Hydrogen Ionization

• Particle equilibrium in a star is dominated by ionized hydrogen.

• Equilibrium is a balance of chemical potentials.

n = 1

n = 2

n = 3

peH n

p = p2/2m

n

n

n

H

QpHHn n

ngkTcmH ln2

p

Qppp n

ngkTcmp ln2

e

Qeee n

ngkTcme ln2

Saha Equation

• The masses in H are related.

– Small amount n for degeneracy

• Protons and electrons each have half spin, gs = 2.

– H has multiple states.

• The concentration relation is the Saha equation.

– Absorption lines

kT

Qe

n

pe

n nen

g

nn

Hn )(

nepH cmcmcmn

222

24)( ngggHg penn

Spectral Types

• The types of spectra were originally classified only by hydrogen absorption, labeled A, B, C, …, P.

• Understanding other elements’ lines allowed the spectra to be ordered by temperature.

• O, B, A, F, G, K, M

• Oh, Be A Fine Guy/Girl, Kiss Me

• Our Brother Andy Found Green Killer Martians.

•

• Type Temperature

O 35,000 K

B20,000 K

A 10,000 K

F 7,000 K

G 6,000 K

K 4,000 K

M 3,000 K

Spectral Classes

• Some bright stars (class):– Sun G2– Sirius A1– Alpha Centauri G2– Capella G8– Rigel B8– Betelgeuse M1– Aldebaran

K5

• Temperature and luminosity are not the same thing.

• Detailed measurements of spectra permit detailed classes.

• Each type is split into 10 classes from 0 (hot) to 9 (cool).

Filters

• Filters are used to select a restricted bandwidth.

– Wide: ~ 100 nm

– Intermediate: ~ 10 nm

– Narrow: < 1 nm

• A standard set of optical filters dates to the 1950’s

– U (ultraviolet – violet): p = 365 nm, = 70 nm

– B (photographic): p = 440 nm, = 100 nm

– V (visual): p = 550 nm, = 90 nm

Filter Sets

• Other filter sets are based on a specific telescope.

– HST: 336, 439, 450, 555, 675, 814 nm

– SDSS: 358, 490, 626, 767, 907 nm

• The standard intermediate filter set is by Strömgren.

– u, b, v, y,

– w: p =486 nm, =15 nm

• CCDs have are good in IR, so filter sets have moved into IR as well.

– U, B, V, R, I, Z, J, H, K, L, M.

– Example M : p = 4750 nm, = 460 nm

Color Index

• The Planck formula at relates the intensity to the temperature.

– Approximate for T < 104 K

• Two magnitude measurements at different temperatures can determine the temperature.

– Standard with B and V filters

– Good from 4,000 to 10,000 K

kThcehc

TW

/5

22),(

T

TVB VBexplog5.2 10

K1065.0 4 k

hc

k

hcT

VBVB

71.0)(

K7090

VBT

Stellar Relations

• The luminosity of a star should be related to the temperature.

– Blackbody formula

– Depends on radius

• Some bright stars:

– Sun G2 4.8

– Sirius A1 1.4

– Alpha Centauri G24.1

– Capella G8 0.4

– Rigel B8 -7.1

– Betelgeuse M1 -5.6

– Aldebaran K5 -0.3

424 TRL

Luminosity vs. Temperature

• Most stars show a relationship between temperature and luminosity.

– Absolute magnitude can replace luminosity.

– Spectral type/class can replace temperature.

-20

-15

-10

-5

0

5

10

15

20

Abs. M

agnitude

O B A F G K MSpectral Type

Sun

Hertzsprung-Russell Diagram

• The chart of the stars’ luminosity vs. temperature is called the Hertzsprung-Russell diagram.

• This is the H-R diagram for hundreds of nearby stars.

– Temperature decreases to the right

Main Sequence

• Most stars are on a line called the main sequence.

• The size is related to temperature and luminosity:

– hot = large radius

– medium = medium radius

– cool = small radius

-20

-15

-10

-5

0

5

10

15

20

Abs. M

agnitude

O B A F G K MSpectral Type

1 solar radius

Sirius

Balmer Jump

• The color indexes can be measured for other pairs of filters.

• The U-B measurement brackets the Balmer line at 364 nm.– Opaque at shorter

wavelength

• This creates a discontinuity in energy measurement.– Greatest at type A– Drop off for B and G

Michael Richmond, RIT

Photometric Comparison

• Stellar classification is aided by different response curves.

Bolometric Magnitude

• Bolometric magnitude measures the total energy emitted at all wavelengths.

– Modeled from blackbody

– Standard filter V

– Zero for main sequence stars at 6500 K

• Luminosity is directly related to absolute bolometric magnitude.

– Flux to apparent bolometric magnitude

Vbol

bol

MMBC

VmBC

bolML 4.028 10W103

bolm4.028 10mW105.2